(Solved): THEOREM 3.4. Let (X,T) be a topological space and let AX. Then the following conditions are ...

THEOREM 3.4. Let $(X,\mathcal{T})$ be a topological space and let $A?X$. Then the following conditions are equivalent. (a) The subspace $\left(\mathrm{A},{\mathcal{T}}_{\mathrm{A}}\right)$ is connected. (b) The set $A$ connot be expressed as the union of two nonempty sets that are separated in $X$. (c) There do not exist $U,V?\mathcal{T}$ such that $U?A?=?,V?A?=?$, $U?V?A=?$, and $A?U?V$.